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No, it was not the first
There are a couple of reasons for initial suspicion of this claim.
Reviews of the episode when it first aired did not comment on the fact that it was the first interracial kiss, nor seem all that bothered by it.
References to it being the first didn't show up until many years later.
So what was the first?
First ever on TV (...

Red shirts have a skewed death rate in TOS due to the nature of their job. The Security role is handled by Ops, which all wear red shirts (Scotty, the engineer, is also Ops). So red shirts die in higher number because Security officers are at a higher risk. They assume guard duties and the dangers inherent to that role.
In TNG and beyond, the color ...

Aircraft carriers dominated battleships in naval fights. But there's reasons for that. The question you need to ask yourself is "would aircraft carriers dominate battleships in space too?"
First, realize that aircraft carriers are no longer the king of naval battles. Missile cruisers are. Missiles have all the reasons why aircraft came to dominate naval ...

I suspect that this started with Roddenberry's vision of the Enterprise.
He had some specific requirements...
"We're [...] out in deep space, on the equivalent of a cruiser-size spaceship. We don't know what the mode of power is, but I don't want to see any trails of fire. No streaks of smoke, no jet intakes, rocket exhaust, or anything like that [.....

They behave like the traditional role of cruisers
A battleship is a dedicated large heavily armed and armored warship designed to work with a larger fleet. It is designed to be a heavy weapons platform (ie. it's got big guns) and take a beating from similar weapons. It requires a larger fleet for fuel, supplies, and auxillary work like scouting, anti-...

Data and Tasha Yar had a drug fueled liaison during season 1 of TNG, The Naked Now. It was never spoken of again, but referenced in Season 2 Measure of a Man.
Season 1 was 2364-2365, First Contact was 2373. Roughly 8 and a half years later. Ergo, he's taking about Natasha.
If Data wasn't incapable of being robotically precise, his keeping to the second ...

There is the TNG episode "The Defector":
In addition to his regular role of Captain Picard, Patrick Stewart also played the role of the holodeck character of "Michael Williams" from Shakespeare's Henry V, Act 4, Scene I, in the opening scenes of the episode.
Stewart requested this out of his love for Shakespeare, himself a member of the Royal ...

It was a mistake.
Out of universe, just a production goof. In universe, Gary Mitchell made a human error.
From Memory Alpha:
The gravestone Mitchell creates for Kirk says "James R. Kirk". According to D.C. Fontana in the introduction for Star Trek: The Classic Episodes 1, when the mistake was discovered, Gene Roddenberry decided that if pressed for an ...

The best I can come up with is Brent Spiner - he played Data on The Next Generation, and appeared onscreen in three episodes as Lore, Data's brother. I'm not sure that this is far enough from the instances of clones or mirrors that you mention, but the characters exist simultaneously and have distinct personalities.
This is the Wikipedia article featuring ...

According to an article about the credits for the movie at TrekMovie.com,
although they [Hemsworth and Morrison] don’t appear in the film, some of their dialog from the 2009 movie can be briefly heard during a dream sequence/flashback.
Presumably that counts as a speaking part, so they have to be in the credits.

In the end it is really up to you. To understand and enjoy the story, all you need are the recent two movies (I'd say).
If you want to understand and notice parallels, nods to the original series, etc. you'd have to watch that as well (including the movies).
But besides that, they did a pretty good job to establish characters, setting etc. without forcing ...

Lt. Nyota Uhura is a communications specialist, with a talent for languages by ear. This extended to her analysis of Krall, as she is the only one he speaks to at length aside from Kirk. Specifically:
The Federation has pushed the frontier for centuries.
But now enough.
This is where it begins, Lieutenant.
This is where the frontier, pushes back.
...

makes no sense as anything intentional.
Well, it could.
The operation being conducted on Sarek is on his heart and involves major blood transfer from Spock.
There is a process for sealing of blood vessels called cauterization
The practice was once widespread for treatment of wounds. Its utility before the advent of antibiotics was said to be effective ...

I would suggest the jutland-vs-midway comparison is fundamentally missing the theme of the series and becoming confused because of that.
At heart, Star Trek is about The naval explorer, harking back to Age-of-sail.
One or more tall ships, a close-knit crew working together for years, visiting strange lands and meeting strange people, Pushing back the ...

Well, remember that Trill hosts being super different was part of Dax's whole backstory - it was a pretty huge deal. Curzon Dax was very different from Jadzia Dax. And I'm pretty sure that Jadzia also wasn't a serial-killing psychopath, like Joran Dax. Plus, it was pretty well established in Star Trek: The Next Generation that in a situation like Jadzia's/...

Two reasons come to mind:
In Star Trek: Generations, the engines were hot because of the just completed Battle of Veridian III. In Star Trek: First Contact, the Enterprise had been quietly orbiting Earth for some time. Cold engines are presumably in less need of coolant.
The Enterprise-E was an upgraded ship compared to the Enterprise-D; it is not ...

Advanced Star Trek space suits and parachutes.
They make a "space-jump". One presumes that since they have a term for it, the suits are engineered specifically for that action.
But really..falling is not the same as re-entry.
As the Space Shuttle reenters the atmosphere, it is going fast. Super-super fast. In orbit, this is about 7,700 m/s (from NASA). ...

The very best I can come up with is Brent Spiner.
He played not two but THREE characters in the same scene, in the 77th episode of Star Trek The Next Generation, 'Brothers', where he appears simultaneously as Data, as Data's evil android brother Lore, and as their creator Dr Noonien Soong.
EDIT: William Shatner kept doing it. In season one, he appears in ...

It was a Klingon propaganda device.
The Klingons apparently appreciated the works of Shakespeare so much that they 'retconned' him to be one of their own poets rather than a human from Earth.
From Wikipedia's article on The Klingon Hamlet, sourced to the DVD commentary on the film Star Trek VI: the Undiscovered Country:
The film's director Nicholas ...

Or was there a story that a writer had submitted that had some scary Halloween related elements that they ended up showing during Halloween?
Mostly..but the timing was deliberate.
Information can be found in These Are The Voyages: Season Two by Marc Daniels..
In there he quotes the writer, Robert Bloch..
“They wanted a Halloween story,” Bloch told ...

More details than you could possibly want are available at Memory Alpha.
In 2370, cadet Crusher resigned from Starfleet Academy after The
Traveler - posing as a villager on Dorvan V - guided him through a
vision of his deceased father who told him that his destiny lay
somewhere other than with Starfleet and that he should not follow in
his ...

As pointed out, there doesn't appear to be an official explanation for this. However, due to several television episodes that show Scotty's penchant for drink (alcohol), plus the habit sailors have for getting very drunk on shore leave, it is quite possible that Scotty was simply talking about a hangover that Bones assisted him in shortening.

The in-universe explanation is quite simple: they don't. They're speaking Klingon and the universal translator makes it sound like they're speaking English.
It's basically impossible to tell which characters speak English due to this device, as by the 24th century they're small enough to fit inside a Starfleet Combadge, as Captain Janeway mentions in The ...

The Borg sphere's approach to time travel is described in two lines of the movie:
DATA: Sensors show chronometric particles emanating from the sphere.
PICARD: They're creating a temporal vortex.
The troublesome question of how the Enterprise returns is dealt with just as abruptly
at the end of the film:
LAFORGE: Captain. I've reconfigured our ...

For storytelling reasons, Roddenberry tried to remove all drama from routine operations. The Galactica reboot's storytelling savored its first FTL jump, with the fear, jump prep, fancy key and all that. After that, it was dropped from the storytelling and not re-explained every episode. They could get away with that because Galactica was a serial: writers ...

He didn't, humans have changed faster than our language, as this conversation between Quark and Jake Sisko in DS9 You Are Cordially Invited... shows:
Jake: I sold my first book today.
Quark: Really? How much did you get for it?
Jake: It's just a figure of speech. The Federation News Service is going to publish a book of my stories about life on ...